The square-root transformation (SRT) of a PDF is defined as a function \(h = +\sqrt{f}\)  (we omit the \(+\)  sign hereafter for notational convenience). The inverse mapping is unique and is simply given by \(f = h^2\)  [1]}. The space of SRTs is given by \(\mathcal {H} = \lbrace h : [0,1] \rightarrow \mathbb {R}_{>0} | \int _0^1 h(x)^2 dx = 1 \rbrace \) , which represents the positive orthant of a unit Hilbert sphere [2]}. The \(\mathbb {L}^2\)  Riemannian metric on \(\mathcal {H}\)  is defined as \(\langle \langle \delta h_1 , \delta h_2 \rangle \rangle = \int _0^1 \delta h_1(t)\delta h_2(t) dt\) , where \(\delta h_1,\delta h_2 \in T_h(\mathcal {H})\)  and \(T_{h}(\mathcal {H}) = \lbrace  \delta h : [0,1] \rightarrow \mathbb {R} | \int _0^1 h(x)\delta h(x)dx = 0\rbrace \) . The geodesic paths and lengths can now be analytically computed due to the Riemannian geometry of \(\mathcal {H}\)  equipped with the \(\mathbb {L}^2\)  metric (\(\mathcal {H}\)  is a Riemannian manifold and the \(\mathbb {L}^2\)  metric on \(\mbox{$\mathcal {H}$}\)  corresponds to the F-R metric on \(\mbox{$\mathcal {F}$}\) ). The geodesic distance between \(h_1,h_2\in \mathcal {H}\)  is simply given by \(d(h_1,h_2)=\theta =\cos ^{-1}\big (\int _0^1 h_{1}(x)h_{2}(x) dx\big )\) . Under this setup, the geodesic distance between two densities \(f_1,f_2 \in \mbox{$\mathcal {F}$}\) , represented by their SRTs \(h_1,h_2 \in \mbox{$\mathcal {H}$}\)  is defined as \(d(f_1,f_2)=d(h_1,h_2)\) .
